Dynamics for Linear Fractional Composition Operator on  \(H^{2}(\mathbb {C}_{+})\)

We consider a class of composition operators induced by linear fractional self-maps on the right half-plane \(\mathbb {C}_{+}\) . Such operators are defined on the Hardy-Hilbert space \(H^{2}(\mathbb {C}_{+})\) , and for these we explore some notions of dynamical systems: chaos, expansiveness, hyperbolicity, hypercyclicity and shadowing.